Transcript Graph each complex number.

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Operations with Complex Numbers
Warm Up
Express each number in terms of i.
1.
2.
Find each complex conjugate.
3.
4.
Find each product.
5.
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6.
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Operations with Complex Numbers
Objective
Perform operations with complex
numbers.
Vocabulary
complex plane
absolute value of a complex number
Holt McDougal Algebra 2
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Operations with Complex Numbers
Just as you can
represent real numbers
graphically as points on
a number line, you can
represent complex
numbers in a special
coordinate plane.
The complex plane is a set of coordinate axes in
which the horizontal axis represents real numbers
and the vertical axis represents imaginary numbers.
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Operations with Complex Numbers
Example 1: Graphing Complex Numbers
Graph each complex number.
A. 2 – 3i
B. –1 + 4i
C. 4 + i
D. –i
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Operations with Complex Numbers
Recall that absolute value of a real number is its
distance from 0 on the real axis, which is also a
number line. Similarly, the absolute value of an
imaginary number is its distance from 0 along
the imaginary axis.
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Operations with Complex Numbers
Example 2: Determining the Absolute Value of
Complex Numbers
Find each absolute value.
A. |3 + 5i|
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B. |–13|
C. |–7i|
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Operations with Complex Numbers
Adding and subtracting complex numbers is
similar to adding and subtracting variable
expressions with like terms. Simply combine the
real parts, and combine the imaginary parts.
The set of complex numbers has all the
properties of the set of real numbers. So you
can use the Commutative, Associative, and
Distributive Properties to simplify complex
number expressions.
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Operations with Complex Numbers
Example 3A: Adding and Subtracting Complex
Numbers
Add or subtract. Write the result in the form
a + bi.
(4 + 2i) + (–6 – 7i)
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(5 –2i) – (–2 –3i)
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Operations with Complex Numbers
You can multiply complex numbers by using
the Distributive Property and treating the
imaginary parts as like terms. Simplify by
using the fact i2 = –1.
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Operations with Complex Numbers
Example 5A: Multiplying Complex Numbers
Multiply. Write the result in the form a + bi.
–2i(2 – 4i)
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(3 + 6i)(4 – i)
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Operations with Complex Numbers
Example 5C: Multiplying Complex Numbers
Multiply. Write the result in the form a + bi.
(2 + 9i)(2 – 9i)
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(–5i)(6i)
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Operations with Complex Numbers
The imaginary unit i can be raised to higher powers
as shown below.
Helpful Hint
Notice the repeating pattern in each row of the
table. The pattern allows you to express any
power of i as one of four possible values: i, –1,
–i, or 1.
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Operations with Complex Numbers
Example 6: Evaluating Powers of i
Simplify
–6i14
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i63
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Operations with Complex Numbers
Recall that expressions in simplest form cannot have
square roots in the denominator. Because the
imaginary unit represents a square root, you must
rationalize any denominator that contains an
imaginary unit. To do this, multiply the numerator
and denominator by the complex conjugate of the
denominator.
Helpful Hint
The complex conjugate of a complex number
a + bi is a – bi.
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Operations with Complex Numbers
Example 7: Dividing Complex Numbers
Simplify.
Pg 130 40 – 102 even, (omit 52,54)105 - 108
Holt McDougal Algebra 2